It is not irrational for President Donald Trump to be seeking a strategic alliance with the Russian government against ISIS, a former CIA official told CNBC on Tuesday.

In an interview on "Squawk Box," Chad Sweet, who served in the intelligence agency's National Clandestine Service and was a Department of Homeland Security chief of staff under the George W. Bush administration, said the Trump administration could work with Russia the same way it aligned with China to counter the threat from North Korea.

"It is absolutely in our interest to have the Russians align with us," said Sweet, co-founder and CEO of global advisory firm and investment bank The Chertoff Group. "It is entirely appropriate for the president to be discussing this common threat against both of our countries."

Sweet's comment came a day after The Washington Post reported that Trump divulged highly classified information about the Islamic State during his meeting with Russian officials last week. Officials told the Post that the information was incredibly sensitive and that its exposure endangers the relationship with an ally, which had not approved sharing the information with Russia.

Secretary of State Rex Tillerson disputed the report and said that during Trump's meeting with Russian Foreign Minister Sergey Lavrov "the nature of specific threats were discussed, but they did not discuss sources, methods or military operations."

Republicans and Democrats expressed concern about the report, however. Republican Sen. Bob Corker of Tennessee said he was worried about a White House "on a downward spiral." Democratic Rep. Al Green called for Trump's impeachment.

Trump responded to the report on Tuesday, saying "As President I wanted to share with Russia."